Begin saying L'David Adonai ori v'yish'i (Psalm 27) on the first day (at night) of Elul, not the first day of Rosh Chodesh Elul (the 30th of Av). The last day to say this Psalm is 22nd of Tishrei (Shmini Atzeret). This Psalm is recited twice daily, in shacharit after shir shel yom and in ma'ariv after alenu, including on Shabbat and Jewish festivals.